#Euclidean construction

[charliecraftsthings]

My first ever attempt at sashiko!  Natural linen thread on denim jeans (left leg), shippo tsunagi pattern.

I thought this pattern would be a good beginner choice, but I was ambitious. I didn't care strongly about making even stitches, I just wanted to try this out.  

I created the grid circle-by-circle (compass and straightedge), using a very cheap drawing compass with a pen taped to it. This turned out to be a TERRIBLE mistake: 

Fabric stretches as you drag a pen over it (Euclidean construction on not-very-Euclidean surfaces, a bad idea--who knew)

As you move farther from the origin, the grid becomes more and more distorted as the errors compound

Since the pants are intact, placing rulers over seams (bumpy!) will distort measurements.

I also didn't take into consideration that at the intersections, I'd have to choose how threads would overlap. So there are some messy intersections.

But I sure learned a lot from this experience! 😅

[spoopy-moose]

So sad that Kentucky route zero never took off on tumblr because it’s so hard to fandomfy such a work because I have so many words of analysis on this game I love it so much

[wrong-directions]

Non-non-Euclidean Biblically Accurate Platonic Solid

[warren-is-dead]

I wanted to see if someone had made a gingerbread R’lyeh, and apparently someone did twelve years ago! Unfortunately their account is deleted so I can’t access the video :’(

[haggishlyhagging]

It took about two hours for Daina Taimina to find the solution that had eluded mathematicians for over a century. It was 1997, and the Latvian mathematician was participating in a geometry workshop at Cornell University. David Henderson, the professor leading the workshop, was modelling a hyperbolic plane constructed out of thin, circular strips of paper taped together. 'It was disgusting,' laughed Taimina in an interview.

A hyperbolic plane is 'the geometric opposite' of a sphere, explains Henderson in an interview with arts and culture magazine Cabinet. 'On a sphere, the surface curves in on itself and is closed. A hyperbolic plane is a surface in which the space curves away from itself at every point.' It exists in nature in ruffled lettuce leaves, in coral leaf, in sea slugs, in cancer cells. Hyperbolic geometry is used by statisticians when they work with multidimensional data, by Pixar animators when they want to simulate realistic cloth, by auto-industry engineers to design aerodynamic cars, by acoustic engineers to design concert halls. It's the foundation of the theory of relativity, and thus the closest thing we have to an understanding of the shape of the universe. In short, hyperbolic space is a pretty big deal.

But for thousands of years, hyperbolic space didn't exist. At least it didn't according to mathematicians, who believed that there were only two types of space: Euclidean, or flat space, like a table, and spherical space, like a ball. In the nineteenth century, hyperbolic space was discovered - but only in principle. And although mathematicians tried for over a century to find a way to successfully represent this space physically, no one managed it - until Taimina attended that workshop at Cornell. Because as well as being a professor of mathematics, Taimina also liked to crochet.

Taimina learnt to crochet as a schoolgirl. Growing up in Latvia, part of the former Soviet Union, 'you fix your own car, you fix your own faucet - anything', she explains. 'When I was growing up, knitting or any other handiwork meant you could make a dress or a sweater different from everybody else's.' But while she had always seen patterns and algorithms in knitting and crochet, Taimina had never connected this traditional, domestic, feminine skill with her professional work in maths. Until that workshop in 1997. When she saw the battered paper approximation Henderson was using to explain hyperbolic space, she realised: I can make this out of crochet.

And so that's what she did. She spent her summer 'crocheting a classroom set of hyperbolic forms' by the swimming pool. 'People walked by, and they asked me, "What are you doing?" And I answered, "Oh, I'm crocheting the hyperbolic plane."' She has now created hundreds of models and explains that in the process of making them 'you get a very concrete sense of the space expanding exponentially. The first rows take no time but the later rows can take literally hours, they have so many stitches. You get a visceral sense of what "hyperbolic" really means.' Just looking at her models did the same for others: in an interview with the New York Times Taimina recalled a professor who had taught hyperbolic space for years seeing one and saying, 'Oh, so that's how they look.' Now her creations are the standard model for explaining hyperbolic space.

-Caroline Criado Perez, Invisible Women

Photo credit

[fjordfocused]

and if nothing else, my liege, I hope my non-euclidean vocabulary fucked your sentence construction to smitherbeans

[klavierpanda]

🌻 :3

I will tell you& about a cool topology fact that uses one of my favourite theorems!

First, a primer of finitely presented groups:

Given a finite set with n elements S={a₁,...,aₙ}, we define a word to be a finite concatenation of elements in S. For example, a₁a₇aₙ is a word. We define the empty word e to be the word containing no elements of S. We also define the formal inverse of the element aᵢ in S, written aᵢ⁻¹, to be the word such that aᵢaᵢ⁻¹=e=aᵢ⁻¹aᵢ, for all 1≤i≤n.

We define the set ⟨S⟩ to be the collection of all words generated by elements of S and their formal inverses. If we consider concatenation to be a binary operation on ⟨S⟩, then we have made a group. This is the free group generated by S, and is called the free group generated by n elements.

Some notation: if a word contains multiple of the same element consecutively, then we use exponents as short hand. For example, the word babbcb⁻¹ is shortened to bab²cb⁻¹.

Note: concatenation is not commutative. So ab and ba are different words!

We now define a relation on the set ⟨S⟩ to be a particular equality that we want to be true. For example, if we wanted to make the elements a and b commute, we include the relation ab=ba. This is equivalent to aba⁻¹b⁻¹=e. In fact, any relation can be written as some word equal to the empty word. In this way, we can view a relation as a word in ⟨S⟩. So we collect any relations on ⟨S⟩ in the set R.

Finally, we define the group ⟨S|R⟩ to be the group of words generated by S subject to the relations in R. This is called a group presentation. An example is ⟨z,z²⟩, which is isomorphic to the integers modulo 2 with addition ℤ/2.

If both S and R are finite, we say that ⟨S|R⟩ is a finite group presentation. If a group G is isomorphic to a finite group presentation we say G is a finitely presented group. It is worth noting that group presentation is by no means unique so as long as there is one finite group presentation of G, we are good.

In general, determining whether two group presentations is really really hard. There is no general algorithm for doing so.

Lots of very familiar groups of finitely presented. Every finite group is finitely presented. The addative group of integers is finitely presented (this is actually just the free group generated by one element).

Now for the cool topology fact:

Given a finitely presented group G, there exists a topological space X such that the fundamental group of X is isomorphic to G, i.e. π₁(X)≅G. This result is proved using van Kampen's Theorem which tells you what happens to the fundamental group when you glue two spaces together.

The proof involves first constructing a space whose fundamental group is the free group of n elements, which is done inductively by gluing n loops together at a single shared basepoint. Each loop represents one of the generators. Then words are represented by (homotopy classes of) loops in the space. Then we use van Kampen's Theorem to add a relation to the fundamental group by gluing a disc to the space identifying the boundary of the disc to the loop in the space that represents the word for the relation we want. We do this until we have added all of the relations we want to get G.

We can do a somewhat similar process to show that any finitely presented group is the fundamental group of some 4-manifold (a space that locally looks like 4-dimensional Euclidean space, the same way a sphere locally looks like a plane). This means that determining whether two 4-manifolds are homeomorphic or not using their fundamental groups is really hard in general because distinguishing finitely generated groups is hard in general.

P.s. I also want to tell you that you're really wonderful :3 <2

[rlyehtaxidermist]

a short guide

euclidean:

a line segment may be constructed between any two points

a line segment may be extended infinitely

a circle may be made centred on any point with any radius

all right angles are equal

if a line crosses two other lines such that the angles they form on one side are less than right angles, the crossed lines intersect on that side

not euclidean:

the surfaces of any finite object

city blocks

pretty much every way you've actually interacted with notions of distance in your life

[lipshits-continuous]

Topological Spaces 2: Topologies and Continuity

Welcome to the second post in my introductory series to topology! The goal today is to give the definition of both topological spaces and continuous maps between them. These definitions will be very familiar from the last post so I highly recommend reading it here.

I do make brief reference to some constructions from linear algebra and abstract algebra however these aren't necessary for understanding the post. They are merely inserted for futher emphasis for those who are familiar.

2.1: Topological Spaces

In the last post, we saw that open sets are potentially a structure that we wish to study. Our definition will be in terms of properties we wish open sets to have without invoking any notion of a distance between point. Fortunately, open sets in metric spaces have properties which fit this exactly in the form of Lemma 1.11. We shall use these to give our definition of a topological space!

Definition 2.1:

Remarks:

We can show by induction that finite intersections of open sets are open. This property is equivalent to (T3), but (T3) is usually an easier property to check.

We can have different topologies on the same set as we shall see in the examples!

Examples 2.2:

Remarks:

The topology on ℝ induced by the absolute value metric is called the standard topology on ℝ. Similarly, The topology induced by the Euclidean norm on ℝⁿ is called the standard topology on ℝⁿ.

The discrete topology is actually the same topology as the topology induced by the discrete metric. The proof involves showing that every singleton set (set that contains one element) is open in X with the discrete metric. Then use Lemma 1.11 to conclude that every set is open in X with the discrete metric.

Just like how some metrics can't be realised as the metric induced by a norm (and how some norms can't be realised as the norm induced by an inner product), not all topologies can be realised as the topology induced by a metric. A topology which can realised as the topology induced by a metric is called metrizable. The area of study dedicated to classifying metrizable spaces is called metrization theory and is very rich (and very much beyond the scope of these posts)

Both the indiscrete topology and the topology in 4) are non-metrizable. We shall see a proof of this in a future post.

One thing we may want to do is to make new topological spaces out of old ones. We don't yet have enough theory developed to do much but one easy construction we may do is putting a topology on subsets of our topology space:

Proposition 2.3:

2.2 Continuous Maps:

Now we have defined what a topology is, we may finally generalise the notion of continuity! Our guide will be Lemma 1.12.

Definition 2.4:

Remarks:

Continuity very much depends on the topology.

This really is a generalisation of continuity since Lemma 1.12 says that for metric spaces this notion is equivalent to our ε-δ definition (Definition 1.3)

We shall now see some examples in the form of the following proposition:

Proposition 2.5:

Now we shall define a special kind of continuous map: homeomorphisms. You may have heard of these from pop maths, especially in the context of a coffee mug and a donut being "the same" to a topologist.

Definition 2.6:

Remarks:

Not only are homeomorphisms bijections between the sets, they also induce a bijection between the topologies:

This means that in some sense the topologies on X and Y are the same!

More formally, one can show that ≅ is an equivalence relation on topological spaces. The proof relies on 1) and 2) from Proposition 2.5

Homeomorphisms are in this sense "topological space isomorphisms" since they preserve the important structure of topological spaces: the topology. This is analagous to how bijective linear maps are vector space isomorphisms and how bijective group homomorphisms are group isomorphisms.

Example 2.7:

In fact, we may show any open interval (a,b) is homeomorphic to ℝ by showing that (0,1) is homeomorphic to (a,b) via an affine map then using the fact that "being homeomorphic" is an equivalence relation.

One major goal of topology is to classify topological spaces up to homeomorphism. This is insurmountable if attempted all at once, even if one restricts to special topological spaces, and is still an active area of research. What is a more approachable goal is to find certain properties that topological spaces have which are invariant under homeomorphism. A vast majority of what follows in this series of posts can be seen as a hunt for these invariants whilst also generalising a lot of familiar concepts we have on ℝ with its standard topology.

[yamada-ryo]

Maybe if I skip the conditioner my hair won't turn into a topologically impossible non-euclidean construct this time

[charliecraftsthings]

My second attempt at sashiko.  Natural linen thread on denim jeans (the same pants), shippo tsunagi pattern.  

Hey so remember how I said creating a grid by circle-by-circle on fabric was a terrible idea?  I decided to do it again anyway 🤦

On this leg, I didn't make the edges of my design straight; I wanted it to look like circles were almost randomly selected to be in the pattern or not.  I also added colourwork, because, again: #ambitiousbeginner 

One of the circles lies with one half on the front of the pant leg, and the other on the back.  I wanted to make it seem like the pattern was continuing around the back of the leg.  I really like how it looks!

[janmisali]

Here is a tiny fraction of the reasons why 2 is special and why 2 MUST WIN:

Euler’s Formula about planar graphs has the constant 2

2 appears in formulas everywhere, especially around tau and pi

2!=2

2+2=2*2=2^2=2^^2=...

Normal distribution has z^2/2 in it

First prime number

truth values have 2 possibilities

Powerset has size 2^n

Infinitely many platonic solids in dimension 2

Completing the square is sososo important

Conic sections are the most complicated well behaved curves, and they come from order 2 polynomials

Lorentz transform is 2ish

F=1/r^2 makes planets travel in ellipse

x.dx=0 -> |x|^2=r^2

Mandelbrot set is all about raising things to the power of 2. And things escape the set if their modulus goes > 2

The sum of inverse powers of 2 is 2.

Groups come from 2-ary operations

Asymmetric objects have 2 possible chiralities

= is a 2-ary relation

Highest order of differential equation that cannot be chaotic

Can split any angle into 2, or any line segment into 2 with simple construction. Can construct sqare root, but not any other roots.

2 used to prove bound on iterated totient function

Biggest group with trivial automorphism group

Dimension of C over R

Antipodal map is null-homotopic in iff the sphere's dimension is not a multiple of 2

Highest moment needed for CLT

Every nontrivial finite degree subfield of an algebraically closed field is degree 2

Fermat primes come from the number 2

Wilson’s theorem is about elements of order 2

Every element order 2 implies Abelian and vector space

Reflection is order 2 symmetry

Every ring except in characteristic 2 has 2nd root of unity

The Euclidean norm is the only norm with continuous rotational symmetry, and it's analytic, with derivative 2x. It also gives the circle parameterised by e^i\theta

It's the dimension in which shapes first appear

It's the smallest possible number base

.

[apocalypticavolition]

Let's (re)Read The Great Hunt! Chapter 37: What Might Be

Everyone has that moment that they've looked back on and wonder, "What if I'd done it differently?" Sometimes we can move on, put what-ifs to rest, but sometimes that question will haunt us for the rest of our lives. Don't let this post be that moment. Don't let future you think to themself, an hour from now, "What if I just hadn't spoiled the whole Wheel of Time series for myself by reading this post? What if I'd enjoyed the books by reading them without spoilers?" Don't click "Keep reading" unless you already know all the spoilers and thus will not forever be wondering.

(And yet... perhaps if you do not choose to spoil yourself, years from now you will look back on this moment and wonder what might have happened if you had.)

This chapter has a Portal Stone icon because we're doing one of the best chapters in the whole damn series.

“We stood it upright,” Alar said, “when we found it many years ago, but we did not move it. It . . . seemed to . . . resist being moved.”

Probably the Stones are all entangled in some sort of higher dimensional quantum process, on one part just to be able to function at all and on another to ensure that no one warps off to an alternate world where it turns out that the Stone was tossed into a volcanic caldera a few years back and thus they instantly die. I wonder if there's wiggle room to allow stuff like Stones falling in one world but not another, or if they're all so tied up that reorienting it in one world caused it to be fixed in all of the others too.

Forgive us for our lack of ceremony in leaving you, but the Wheel waits for no woman.

Since I've given Jordan a bit of guff here and there for enforcing his own gender beliefs on the settings, point here for having Verin use "woman" as the default term. Hashtag HER-story, amirite?

Ingtar’s back stiffened. “I hold back at nothing. Take us to Toman Head or take us to Shayol Ghul. If the Horn of Valere lies at the end, I will follow you.”

Really you might argue that this here was the real moment of redemption for Ingtar and that all the rest is just the formality of seeing it through.

I have never used a Stone; that is why your use is more recent than mine.

"Bitch I'm just covering so no one has to know you're the Dragon Reborn. Do the plot thing already!"

Also I kinda feel that Verin is really stretching the oaths she's pretending to have here.

What would I not give to talk with this girl of yours? Or better, to put my hands on her book. It is generally thought that no copy of Mirrors of the Wheel survived the Breaking whole. Serafelle always tells me there are more books that we believe lost than I could credit waiting to be found.

Honestly, even though rumors are so rarely right in this world, I think popular opinion is correct and none survive. I also find it pretty doubtful that there's that many lost books left to be recovered at this point: three and a half thousand years is a long-ass time, too long for most forms of writing to survive.

Apparently, not every Stone connects to every world, and the Aes Sedai of the Age of Legends believed that there were possible worlds no Stones at all touched.

Among others, any timeline where a Portal Stone was never constructed would by definition remain off of the network. I wonder how they decide what Stones connect to what, though. Was it perhaps based on what ifs related to the nearby areas? What other worlds were missing?

With one finger she outlined a rectangle containing eight carvings that were much alike, a circle and an arrow, but in half the arrow was contained inside the circle, while in the others the point pierced the circle through. The arrows pointed left, right, up and down, and surrounding each circle was a different line of what Rand was sure was script, though in no language he knew, all curving lines that suddenly became jagged hooks, then flowed on again.

I expect that these worlds have extremely non-Euclidean geometries at play, based on how they were used to make the Ways. Likely the arrow has something to do with a physical force, probably gravity.

As my father would have said, it’s time to roll the dice.

Headcanon: Mat is Verin's dad reborn.

“I am Rand al’Thor,” he growled. “I am not the Dragon Reborn. I won’t be a false Dragon.” “You are what you are. Will you choose, or will you stand here until your friend dies?”

As I've said before, the one thing Rand's not allowed to do is stand still: every time he does the pressure only mounts until he has to act. Verin at least offers the kindness of spelling it out for him.

The flame consumed fear and passion and was gone almost before he thought to form it. Gone, leaving only emptiness, and shining saidin, sickening, tantalizing, stomach-turning, seductive. He . . . reached for it . . . and it filled him, made him alive. He did not move a muscle, but he felt as if he were quivering with the rush of the One Power into him.

After all this time, I still can't decide if being a channeler would be really awesome or really awful.

“Father!” Rand screamed. Clawing his belt knife from its sheath, he threw himself over the table to help his father, and screamed again as the first sword ran through his chest.

Though of course the Mirror Worlds take from the Many Worlds Theory, we must remember that they're not actually the same. The Many Worlds Theory is a way of resolving one of the fundamental mysteries of quantum mechanics. When not observed, particles don't have discrete locations but probabilites of being here, there, or even over there. These odds are called a "waveform". When observed, the waveform collapses and the particle is only in one of those places. The thing scientists don't get yet is the mechanic of that collapse nor the reason. Many Worlds Theory says "The collapse is an illusion. All of those possibilities exist somewhere but since we can only exist in one place we can only ever observe one possibility. All worlds continue on, none with more value or reality than any other except in that those who exist in only one must favor where they are."

This is not what the Mirror Worlds are. The Pattern of Ages is a specific framework which dictates one reality (T'A'R) reigns supreme above all the others, and that among these the closest reflection (the Prime Reality) is inherently more valid than the increasingly distorted copies.

In Many Worlds theory, one can discuss the relative probabilties of different timelines. One location for a particle might have had a 2/3rds chance of being the real one while the other two were each only a sixth. Amid the Mirror Worlds, there's no such thing. T'A'R and the Prime reality each have a 100% chance of being true and all other worlds have a 0% chance of happening.

That said, the Wheel does seem to think some Mirror Worlds are more plausible than others, and I think Rand's journey is - at least at first - moving in order of descending plausibility. Him dying immediately when the story began is a very "likely" outcome - to some degree more likely than other potential deaths later in the timeline just because in each of those scenarios Rand had a little more experience to keep him going.

There was a year when neither merchants nor peddlers came, and when they returned the next they brought word that Artur Hawkwing’s armies had come back, or their descendants, at least.

It's bizarrely heartening to think that even the Seanchan invasion will completely miss that the Two Rivers exists.

Also note that this world - where Rand is never found by Fain or Moiraine and never leaves as a result - seems next most likely amid the categories.

Egwene grew frightened when the moods were on him, for strange things sometimes happened when he was at his bleakest—lightning storms she had not heard listening to the wind, wildfires in the forest—but she loved him and cared for him and kept him sane, though some muttered that Rand al’Thor was crazy and dangerous.

I wonder what happened to this Egwene that she accepts the Two Rivers life without complaint while Rand is forever ranting about how life should be. I also do think that the haters should remember that this is the "no inciting incident" default Egwene: a caring person who stays with Rand until the end. The pair grow apart because of outside forces, not because Egwene is fundamentally flawed as a person.

Women came, too, shouldering what weapons they could find, marching alongside the men. Some laughed, saying that they had the strange feeling they had done this before.

This is both nice foreshadowing for how the Two Rivers folk will respond to the real Shadowspawn invasion and another hint of the old blood amid the people. It would not be surprising at all if many of them were truly the last of Manetheren reborn.

Tam tried to console Rand when Egwene took sick and died just a week before their wedding.

The nextmost implausible sort of world: no inciting incident and Rand survives his channeling sickness but Egwene does not. Being a slightly mainer character than she is, it tracks that this is more plausible than a world where he dies young while she stays on track to be Wisdom.

Elayne did not look at him, of course; she married a Tairen prince, though she did not seem happy in it.

I'd be upset too in this position. What a strange world this is, that a gal who should be the first Aes Sedai queen in centuries should end up married to anyone from Tear. What the fuck is going on at the White Tower to lead to this? I would guess that the reason Moiraine didn't find Rand is that Siuan isn't Amyrlin and that whoever is in charge instead has run the place into the ground.

Also, assuming "prince" means "son of a High Lord or Lady", if not "High Lord" directly, I wonder which horrible family Elayne is stuck with.

He knew he was mad, and did not care. A wasting sickness came on him, and he did not care about that, either, and neither did anyone else, for word had come that Artur Hawkwing’s armies had returned to reclaim the land.

1. It seems that this Rand is doomed to never be able to complete his character development without the actual plot happening.

2. What's delayed the Seanchan by years if not a decade? How far back does this timeline's divergence have to be to account for all of this?

Many of the people of Caemlyn had fled already, and many counseled the army to retreat further, but Elayne was Queen, now, and vowed she would not leave Caemlyn. She would not look at his ruined face, scarred by his sickness, but he could not leave her, and so what was left of the Queen’s Guards prepared to defend the Queen while her people ran.

I expect that this was foreshadowing Caemlyn's importance in the Last Battle, an importance that Sanderson didn't fully follow up on. Even in this life, Rand finds himself head of an army by Elayne's side leading a desperate last stand.

I have won again, Lews Therin. Flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker flicker.

There's some great details in the paragraph that precedes this bit, much too much to analyze. It says a great deal about the central nature of Rand to the Pattern that he can have so many bizarre outcomes: I expect no one else in the party had anywhere near so much variety in their lives.

We also get confirmation that Rand is Aiel, which is nice, though it's sad that the closest thing to a reference Aviendha gets in this procession is "women he had never seen before".

Of course, Rand's lovers aren't the important part here. Elayne and Min get mentioned but it is Egwene who receives a similar multi-faceted fate. So often she is a central figure in his life; she can't help but be his opposite even when their lives have gone horribly off-script.

And of course, our iconic line. The Dark One wins again and again, but like I already said: none of these worlds have even a 0.00000000000000000000000000000001 percent chance of happening. None of these victories matter in the slightest.

“Does it surprise you that your life might go differently if you made different choices, or different things happened to you? Though I never thought I—Well. The important thing is, we are here. Though not as we hoped.”

I desperately wish I had the slightest idea what Verin had seen in her procession. Were there worlds where she avoided the Black Ajah altogether, or worlds where she happily threw in with them? Maybe a world where she poisoned Cadsuane, or one where she was in Moiraine's place and threw Lanfear through the twisted red door?

You should not have tried to bring us directly here. I don’t know what went wrong—I don’t suppose I ever will—but from the trees, I would say it is well into late autumn.

Presumably it's the nature of those arrow worlds. I've joked about the Ways being akin to the inside of a black hole and suggested that they had strange geometry and I expect this is proof. They did come instantly but it also took four months by another spacetime's reckoning.

“Rand, I’d never tell anyone about—about you. I wouldn’t betray you. You have to believe that!”

It's true! Mat doesn't do that in this reality and none of the other ones count. But I do think he was tempted at points. Not enough to go through with it (and he had no real opportunity to do so), but still. Now though, that door is permanently closed.

The curly-haired youth dropped his hands from his face with a sigh. Red marks scored his forehead and cheeks where his nails had dug in. His yellow eyes hid his thoughts.

Wolf boy here probably had one hell of a time in the pack. Or perhaps he just got out of that weird timeline where he mistakes Laila for a Trolloc.

Rand backed away when she reached for him. “Don’t be foolish,” she told him. “I don’t want your help,” he said quietly. “Or any Aes Sedai help.” Her lips twitched. “As you wish.”

1. I expect that nearly everyone has now forgotten a good deal of the experience thanks to Verin's help, which ironically helps Rand even though he doesn't want it.

2. Verin must really chafe at the sheer ingratitude of this, considering just how much she's doing for the dumb boy.

3. That's the end of our chapter folks! Next time: Remember Egwene? She still exists!

[desceros]

hi, just gonna start this off by saying I'm in love with your stories!! Like Dood, I've read your stuff an unhealthy amount of times. I was just wonderin'... *twiddles fingers* if you ever get time, could you drop some tips on how you write your stories? And what I mean by that is, like how you develop your main character and their relationship with the turts, then how you plan your story without getting stuck? Because your stories are flawless and I always have trouble getting past plot holes! (btw, sorry if this was a lot! I completely understand if you don't have time to answer this!)

uwahh, thank you so much!! i love hearing that people enjoy my writing, heehee = v= and i'm also a sucker for this kind of question so thanks for giving me an excuse to talk about it :D

sooo usually i start with a plot nugget (e.g. "a meet cute at the aquarium where you see someone cute through the tank but don't catch their name"). then i think ok, who does that mesh with, pairing wise? for that fic, euclidean line, i wanted to write rise donnie and he fit the prompt very well, so that settles that. sometimes it's 'oh this prompt begs for this character, so i'm going to be writing him now i guess.' just depends.

with the plot nugget usually comes a vibe, and that vibe kind of dictates the mood of the fic. sometimes my fics are humorous (think goldilocks), and sometimes they're more somber (think EL). with that vibe comes the, uh, "mood" of the reader insert character. since i knew EL was going to be a study of donnie's emotional fallout with krang, i wanted to have a reader who was good at dealing with emotions to be a kind of guiding light for him. goldilocks has a chaotic reader to match the chaotic feel of that fic. etc. that gives me a feel for what the reader's personality is going to be, and from there, i can construct other pieces of things like how they react with the other turtles. leonardo is going to react differently to a sassy reader dating raph than he is to an emotionally-sensitive reader dating donnie, after all, so their relationships kind of fall into place once i know who the reader is.

i like to have a "best friend" character for the reader outside of the romantic relationship. it helps the reader feels like a person with a life outside of the romantic relationship, and it also gives them a chance to have conversations that drive a plot beyond just "when are the main characters gonna smooch". not that that kind of thing isn't interesting (i'm a romance writer for god's sake!), it's just not the only thing i like to have going on. so sometimes i pick the best friend at random (e.g. in amaranthine, i wanted to practice writing raph so i made him the best friend character), and sometimes it's pretty important to the plot (e.g. in the tea fic where splinter being the bestie is actually pretty important to the emotional core of the fic). i match the reader's personality to the best friend character as well, and this helps me create a multi-dimensional person. like... what kind of person would leo pine after, but who is also drinking tea all the time with splinter? what kind of person would donnie want to explore pain kink with, but is also best friends with raph? that kind of thing.

as for the plot and how to avoid plot holes, what i like to do is have a connecting thread that weaves through the entire fic. going back to euclidean line, that fic was all about the jellyfish. you and donnie meet at the jellyfish tank because the two of you are drawn to them, comforted by your different traumas by the idea of existing in the life of a jellyfish. so then you take all of the events of the fic and you pin them to that thread. so little things like... meeting at the jellyfish tank. the smut making them feel like they're floating like a jellyfish. painting a jellyfish for his lab. it gives your fic a sense of continuity that ties everything together, and also gives you an emotionally cathartic line to end on. euclidean line ends with the two of you feeling like jellyfish together. the tea fic ends with leo saying your tea is better than splinter's. the bruise fic ends with the same 'donnie thinks about it a lot' line from the opening. it's a kind of... cheat sheet for satisfying bookends.

from there you just expand out and add plot lines one by one. euclidean line had the romance plot line, but also the meeting the hamato clan plot line. it's the same idea with that one with the tension of you the reader knowing that these are donnie's family, but the insert-chan not knowing. they all build on the same idea, creating continuity through the fic.

then, at the end when you finish your first draft, you go through and you just make sure all the little loops are closed. sometimes i'll catch things that i included at the beginning and either take them out because they didn't go anywhere, or i expand on them a little so that they end in a nice bowtie with everything else. eventually you get to where you just kind of... keep all of them juggling in your head and weave them in organically (which is how you end up chronically writing 30k one shots, oops).

WOW that got super long, but i hope i answered your question?? if not you can ask something more specific and i'll try to dig in but this is probably a pretty good overview, heehee. the best advice i can give to you is just never to give up and always keep writing what YOU want to read, because the love of what you're making is really the thing that'll keep you passionate and excited to go forward. good luck, and i hope this helps you a little! :D

[time-to-kill-god]

Item #: SCP-8106

Object Class: Euclid

Special Containment Procedures:

SCP-8106 is to be contained within a custom-made cell constructed from an anomalous alloy designated as "kronosium." Access to the containment area requires Level 3 authorization or higher.

Description:

SCP-8106 is a humanoid entity with various avian characteristics, including wings, scaly claw-like hands, and feathers. Its body emits a bright, glowing light that makes it difficult to discern details of its physical form. 

SCP-8106 was first encountered by Dr. Jack Bright, who at the time was taunting SCP-682 with a katana and cannabis. SCP-8106 suddenly appeared and forcefully subdued SCP-682, before collapsing from apparent exhaustion.

When awakened by Dr. Bright poking it with the katana, SCP-8106 promptly vanished and was later found in a small hut in Scotland. During interactions with researchers and D-Class personnel, SCP-8106 has displayed a polite, British-accented demeanor, often offering tea before experiencing what has been described as a "glitching out" phenomenon.

This "glitching out" involves SCP-8106 losing its Euclidean form, emitting various colors, and crying. It then proceeds to teleport the affected individual(s) to a location they consider a "safe space." After these incidents, SCP-8106 has apologized and provided instances of SCP-500 or a variant that heals mental issues.

When SCP-8106 was invited to visit the Foundation, it initially appeared confused and content, but upon encountering D-Class personnel, it "glitched out" in a red state, immediately teleporting all D-Class to an unknown location. It then proceeded to harm, but subsequently heal, Foundation personnel who had advocated for less humane treatment of D-Class. 

Attempts to approach the hut in Scotland where SCP-8106 was originally found have resulted in an impenetrable force field surrounding the structure, occasionally delivering electrical shocks to researchers. All photographic and video evidence of the hut has been lost in a mysterious fire.

SCP-8106 has on occasion willingly visited the Foundation, only to attempt to free D-Class personnel and damage the facility, for reasons that remain unclear. During one such visit, SCP-8106 approached the containment unit of SCP-343, resulting in radio signals of a distressed man pleading for mercy. When questioned, SCP-343 cryptically advised the Foundation to "dig down far enough," leading to the discovery of a strange, unidentifiable black substance that caused severe depression and death in those who handled it.

Further testing with the black substance revealed that it does not appear to be composed of molecules or atoms, and its presence seems to be connected to SCP-8106. When researchers attempted to throw the substance at SCP-8106, the entity immediately collapsed and lost consciousness, only to wake up and faint again repeatedly. Upon finally regaining consciousness, SCP-8106 inquired about the "kronosium" used to construct its containment cell, leading to the designation of this material as SCP-8106-1.

Since its containment, SCP-8106 has remained quiet and despondent, staring at the containment cell wall. All D-Class personnel placed within the cell leave in better health, and any attempts to harm SCP-8106 result in the attacker's disappearance.

The anomalous effects surrounding the hut in Scotland have remained active and have been designated as SCP-8106-2.

[noctumsolis]

According to Lee Smolin, the eight stages by which mathematics develop from the study of relations among natural objects are as follows:

1. Exploration of the natural case (e.g., countable objects, 3D spatial geometry, etc.)

2. Formalization of natural knowledge (e.g., arithmetic, trigonometry)

3. Exploration of the formalized natural case.

4. Evocation and study of variations on the natural case (e.g., non-euclidean geometry)

5. Invention of new modes of reasoning (e.g., axioms)

6. Unification of cases within more general frameworks (e.g., Riemannian geometry)

7. Discovery of relationships between constructions generated autonomously within mathematics

8. Discovery of the applicability of nature of knowledge developed internally.

Smolin rejects the idea that the study of mathematics constitutes the exploration of a Platonic realm separate from physical reality. Rather, he thinks that it represents an evocation from reasoning about items in the natural world. Thus, its applicability to physics occurs precisely because it is, historically, rooted in reasoning about the physical universe.

I need to learn more about this. So far it makes perfect sense!